EXAMPLE 1.6.10

Gomer is going to order a frozen tofu cone. The following toppings are available:

granola crumbles

seaweed sprinkles

carob chips

frosted alfalfa sprouts

roasted soybeans

1. How many topping combinations are possible if he will choose either three or four toppings?

A. 50

B. 10

C. 15

D. 21

SOLUTION

C(5,3) + C(5,4) = 10 + 5 = 15

2. How many combinations are possible if he may choose all, some or none of the items?

SOLUTION

We saw earlier (Unit 3 Module 1) that this problem could be solved using the Fundamental Counting Principle, by making 5 yes-or-no decisions; the correct answer was 32.

Another more complicated approach to the correct answer is to recognize that Gomer must choose either 0 toppings, or 1 topping, or two toppings, or 3 toppings, or 4 toppings, or 5 toppings.

0 toppings: C(5,0) = 1

1 topping: C(5,1) = 5

2 toppings: C(5,2) = 10

3 toppings: C(5,3) = 10

4 toppings: C(5,4) = 5

5 toppings: C(5,5) = 1

1 + 5 + 10 + 10 + 5 + 1 = 32 different combinations.